Characters of unipotent groups over finite fields

Date: 03/12/2007

Lecturer(s):

Dmitriy Boyarchenko (University of Chicago)

Location:

University of British Columbia

Topic:

Let G be a connected unipotent group over a finite field
F_q. For each natural number n, we have the unique extension F_{q^n} of
F_q of degree n, and we can form the finite group G(F_{q^n}) of points
of G defined over F_{q^n}. An interesting problem, motivated by
Lusztig's theory of character sheaves, is to study irreducible
characters of these finite groups (over an algebraically closed field
of characteristic 0) and relate them to the geometry of G. If the
nilpotence class of G is less than p (the characteristic of the field
F_q), there exists an explicit description of irreducible characters of
G(F_{q^n}), provided by Kirillov's orbit method. It allows one to
introduce the notion of an L-packet of irreducible representations of
G(F_{q^n}). This notion is morally analogous to the notion of an
L-packet in Lusztig's theory, even though Lusztig's definition cannot
be applied to unipotent groups. If the nilpotence class of G is at
least p, no analogue of the orbit method is known to us. Nevertheless,
we have succeeded in giving a geometric definition of L-packets of
irreducible characters of G(F_{q^n}) for every connected unipotent
group G over F_q. My talk will be devoted to giving a precise statement
of our result, explaining some motivation behind it, and sketching a
few of the essential ideas used in its proof. (Morris): It is known
that finite-index subgroups of the arithmetic group SL(3,Z) are not
left orderable. (In other words, they have no interesting actions on
the real line.) This naturally led to the conjecture that most other
arithmetic groups (of higher real rank) also are not left orderable.
The problem remains open, but joint work with Lucy Lifschitz verifies
the conjecture for many examples, including every finite-index subgroup
of SL(2,Z[sqrt(3)]) or SL(2,Z[1/3]). The proofs are based on the fact,
proved by D.Carter, G.Keller, and E.Paige, that every element of these
groups is a product of a bounded number of elementary matrices.